linear algebra exam 2

An indexed set of vectors B={b1,...,bp} b equals the set {b 1 , b p} in V is a basis for H if

(i) B is a linearly independent set, and (ii) the subspace spanned by B coincides with H; that is, H=Span{b1,...,bp}

Given a specific vector v, how to tell if v is in Nul A.

Just compute A v.

change-of-coordinates matrix from B to the standard basis in ℝn

PB. Left-multiplication by PB p sub b transforms the coordinate vector [x]B left bracket x right bracket sub b into x.

finite-dimensional

If V is spanned by a finite set. and the dimension of V, written as dim V, is the number of vectors in a basis for V.

Find a spanning set for a subspace:

If v 1 ... v p are in a vector space​ V, then Span{ v 1 v p } is a subspace of V. Using the arbitrary vector​ above, W is a subspace of set of real numbers R4.

Find the​ change-of-coordinates matrix from B to the standard basis in set of real numbers R 2

Pb is just the two vectors in a matrix

Given a specific vector v, how to tell if v is in Col A

Row operations on [Av] are required

the subspace spanned by {v1,...,vp}.

Span {v1,...,vp}

change of coordinates matrix from B to C

just take weights in the columns where b=something c and put it into a matrix

statements equivalent to the matrix is invertible

m. The columns of A form a basis of ℝn. blackboard bold cap r to the n , . n. Col A=ℝn cap col eh equals , blackboard bold cap r to the n o. dim Col A=n dim , cap col eh equals n p. rank A=n rank , eh equals n q. Nul A={0} cap nul eh equals the set 0 end set r. dim Nul A=0

what is the basis of row A

non zero rows of row reduced

how to write n in terms of rank and null

rank A (# pivot) + dim Nul A ( # non pivot) = n

Determine whether a set is a basis for R3.

row reduce and see if these are true: a. The matrix A is an invertible matrix. b. The matrix A is row equivalent to the n times n identity matrix. c. The matrix A has n pivot positions. d. The equation Ax=0 has only the trivial solution. e. The columns of A form a linearly independent set. f. The columns of A span R n.

what axioms hold for all u, v, w in V and for all scalars c and d

1. The sum of u and v, denoted by u+v, u plus v comma is in V. 2. u+v=v+u. u plus v equals v plus u . 3. (u+v)+w=u+(v+w). open u plus v close plus w equals u plus open v plus w close . 4. There is a zero vector 0 in V such that u+0=u. u plus 0 equals u . 5. For each u in V, there is a vector −u negative u in V such that u+(−u)=0. u plus open negative u close equals 0 . 6. The scalar multiple of u by c, denoted by c u, is in V. 7. c(u+v)=cu+cv. c open u plus v close equals c u plus c v . 8. (c+d)u=cu+du. open c plus d close u equals c u plus d u . 9. c(du)=(cd)u. c open d u close equals open c d close u . 10. 1u=u.

the coordinate mapping x↦[x]B

is a one-to-one linear transformation from V onto ℝn.

Find a spanning set for a subspace:

Span() denotes the set of all vectors that can be written as linear combinations of v 1 v p. A linear combination is any sum of scalar multiples of vectors.​ Therefore, for one vector v in set of real numbers R cubed​, ​Span{v​} is the set of all vectors that are scalar multiples of v. Write the vectors in H as scalar multiples of a vector v. Since every vector in H can be written as a linear combination, H span

what are kernel and range

The kernel (or null space) of such a T is the set of all u in V such that T(u)=0 (the zero vector in W). The range of T is the set of all vectors in W of the form T(x) for some x in V.

basis for Col A

The pivot columns of a matrix A

row space

The set of all linear combinations of the row vectors Row A is a subspace of ℝn

what is the standard basis for ℝn

The set {e1,...,en}

when nul A = 0

The transformation T Rn → Rm defined by x → Ax is one-to-one. Ax=0 has only the trivial solution the columns of A are linearly independent

Determine whether a vector or matrix is in a given subspace:

To determine if w is in the subspace spanned by ​{v 1​, v 2​, v 3​}, it is necessary to determine if w is a linear combination v 1​, v 2​, and v 3. This is the same as asking if the equation x 1 v 1 + x 2 v 2 + x 3 v 3 =w has a solution.

coordinate vector of x (relative to B ), or the B-coordinate vector of x.

[x]B=⎡⎣⎢⎢⎢c1⋮cn⎤⎦⎥⎥⎥

vector space

a nonempty set of elements, say {u, v, w , ...}, called vectors, on which two operations addition, u + v , and multiplication by a scalar, cu, are defined for any u, v in V and for any scalar c.

isomorphism

a one-to-one linear transformation from a vector space V onto a vector space W is called an isomorphism from V onto W Every vector space calculation in V is accurately reproduced in W, and vice versa.

spanning (or generating) set for H

a set {v1,...,vp} the { v 1 , vp } in H such that H=Span {v1,...,vp}

what is the spanning set theorem

a. If one of the vectors in S—say, vk v sub k —is a linear combination of the remaining vectors in S, then the set formed from S by removing vk v sub k still spans H. b. If H≠{0}, h not equal to the set 0 end set , comma some subset of S is a basis for H.

Demonstrate that a given set is or is not a vector space. If u and v are in​ V, is u+v in​ V? If u is in W and c is any​ scalar, is cu in​ W?

a. The zero vector of V is in H. b. H is closed under vector addition. That​ is, for each u and v in​ H, the sum u+v is in H. c. H is closed under multiplication by scalars. That​ is, for each u in H and each scalar​ c, the vector cu is in H

Determine whether a given set is a subspace: Determine if the given set is a subspace of set of prime numbers P Subscript n. The set of all polynomials in set of prime numbers P Subscript n such that p​(0)=0

a. The zero vector of V is in H. b. H is closed under vector addition. That​ is, for each u and v in​ H, the sum u+v is in H. c. H is closed under multiplication by scalars. That​ is, for each u in H and each scalar​ c, the vector cu is in H

subspace

a. The zero vector of V is in H. b. H is closed under vector addition. That is, for each u and v in H, the sum u+v u plus v is in H. c. H is closed under multiplication by scalars. That is, for each u in H and each scalar c, the vector c u is in H.

null space

is the set of all solutions of the homogeneous equation Ax=0 is a subspace of Rn

Find the coordinate vector xb of x relative to the given basis B

create | bx | augmented matrix and row reduce

Find a spanning set for the null space of the matrix

find the general solution of Ax=0 in terms of free variables Every linear combination of u, v, and w is an element of Nul A and vice versa. Thus {u,v,w} is a spanning set for Nul A.

Find a basis for a null space

first find Nul​ A, of a matrix A is given by the set of all solutions of the equation Ax=0. then to check if basis is linearly independent and spans the null space of the given​ matrix

the set {v1,...,vp} is said to be linearly dependent if

has a nontrivial solution, that is, if there are some weights, c1,...,cp, c sub 1 , comma dot dot dot comma , c sub p , comma not all zero,

A set of vectors is linearly independent if

has only the trivial solution

when is the column space all of Rm?

if and only if the equation Ax=b has a solution for each b in ℝm. or if the linear transformation maps onto

determine is colA =R3

if pivot every row

Determine whether a vector is in the null or column space of a matrix

in nul A if Av=0. in col A if Ab is consistent

Use an inverse matrix to find xb for the given x and b

inverse of Px * x = xb

Find the vector x determined by the given coordinate vector xb and the given basis B.

take thee values of xB times B to find x

Find a Basis for Col A.

the pivot columns of matrix A form a basis for Col A.

column space

the set of all linear combinations of the columns of A Span{a1,...,an} is a subspace of ℝm

coordinates of x relative to the basis B (or the B-coordinates of x)

the weights c1,...,cn such that x=c1b1+⋯+cnbn

how to get xb from P-1 and xc

xb = P-1 * xb

coordinate mapping (determined by B )

x↦[x]B